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Gravity (and Capillary) 
waves
FABIO ROMANELLI
Department of Mathematics & Geosciences 
University of Trieste
romanel@units.it
 
Corso di Laurea in Fisica - UNITS
Istituzioni di Fisica per il Sistema Terra
mailto:romanel@dst.units.it
http://moodle2.units.it/course/view.php?id=7022
ì
Incompressible fluids
In many cases of the flow of fluids their density may be 
supposed invariable, i.e. constant throughout the volume 
and its motion and we speak of incompressible flow:
ρ = constant
Conservation of matter
Euler equation
i.e. the time taken by a sound signal to traverse distances must be small 
compared with that during which flow changes appreciably
∇⋅V = 0
The conditions under which the fluid can be considered incompressible 
are:
Δρ = ΔP
c2
≈ 1
c2
ρ ∂V
∂ t
λ
⎛
⎝⎜
⎞
⎠⎟
≈ 1
c2
ρV
τ
λ
⎛
⎝⎜
⎞
⎠⎟
∂ρ
∂ t
> λ
c
ì
Incompressible & Irrotational flow
From Euler equations we have that only viscosity can generate vorticity 
if none exists initially. And if the flow is irrotational rot(V)=0, and thus
V=grad(𝝓), the flow is called potential.
Euler equation rot (V)=0
and we can separate the variables...
∇2(𝜙)=0
the potential has to satisfy Laplace equation:
Conservation of matter div (V)=0
and if it is also incompressible:
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Separation of variables + BC at bottom
Let us consider a velocity potential propagating along the x-axis and 
uniform in the y- direction: all quantities are independent of y. 
We shall seek a solution which is a simple periodic function of time 
and of the coordinate x, i.e. we put:
φ = F(z)cos(kx −ωt)
d 2F
dz2
− k 2F = 0 F(z) = Aekz + Be-kz⎡⎣ ⎤⎦then
and if the liquid container has depth h, 
there the vertical flow has to be 0:
vz =
dF
dz z=−h
= 0 ⇒ B = e-2kh A
ì
BC at bottom
and this leads to:
Thus, at the bottom (z=-h) the cosh(0)=1, while at top it is 
cosh(kh), thus F grows as z goes from bottom to top values. 
If the container is infinitely deep (h goes to infinity) we 
have that B has to be 0 and the potential as well is going to 0: 
F(z) = 2Ae-kh cosh k(z + h)⎡⎣ ⎤⎦
F(z) = Aekz
ì
Gravity waves
The free surface of a liquid in equilibrium in a 
gravitational field is a plane. 
If, under the action of some external perturbation, the 
surface is moved from its equilibrium position at some 
point, motion will occur in the liquid. 
This motion will be propagated over the whole surface 
in the form of waves, which are called gravity waves, 
since they are due to the action of the gravitational field. 
We shall here consider gravity waves in which the 
velocity of the moving fluid particles is so small that we 
may neglect the term (V•grad)V in comparison with ∂/∂t 
in Euler's equation. 
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Gravity waves
The physical significance of this is easily seen:
During a time interval of the order of the period, 𝜏, of the oscillations of 
the fluid particles in the wave, these particles travel a distance of the order 
of the amplitude, a, of the wave. Their velocity V is therefore of the order 
of a/𝜏. It varies noticeably over time intervals of the order of 𝜏 and 
distances of the order of 𝜆 in the direction of propagation (where 𝜆 is the 
wavelength). Hence the time derivative of the velocity is of the order of 
V/𝜏, and the space derivatives are of the order of V/𝜆. 
i.e. the amplitude of the oscillations in the wave must be small 
compared with the wavelength
Thus the 
condition 
is equivalent to 
(V ⋅ grad)Vand it acts like a long
wave. The vertical displacement peaks at the
ocean surface and drops to zero at the seafloor.
The horizontal displacement is constant through
the ocean column and exceeds the vertical com-
ponent by more than a factor of ten. Every meter
of visible vertical motion in a tsunami of this
frequency involves ≈10m of “invisible” hori-
zontal motion. Because the eigenfunctions of
long waves reach to the seafloor, the velocity of
long waves are sensitive to ocean depth (see top
left-hand side of Fig. 1). As the wave period
slips to 150s (middle Fig. 2), λ decreases to
26km -- a length comparable to the ocean depth.
Long wave characteristics begin to break down,
and horizontal and vertical motions more closely
agree in amplitude. At 50s period (right, Fig. 2)
the waves completely transition to deep water
behavior. Water particles move in circles that
decay exponentially from the surface. The eigen-
functions of short waves do not reach to the sea-
floor, so the velocities of short waves are inde-
pendent of ocean depth (see right hand side of
Fig. 1, top). The failure of short waves (λf
+ ∂φ
∂t z= f
= 0
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Gravity capillary waves dispersion
neglecting gravity in deep water
C
p
2
K min
g/k
C
p
2
min
K
sk/ρ
sk/ρg/k +
Fig 6-2. Effects of gravity and surface tension on wave speed in deep water
One can check that the kmin corresponds to a minimum of Cp. The Cp min can now be computed
by substituting kmin in the expression for Cp:
C2
p min = 2
√
sg/ρ
In the case of clear air-water interface, s= 0.073 [N/m]. Assuming ρ to be 1000 [kg/m3] and g to
me 9.8 [m/s2], we observe that the minimum wave speed on water is 0.23 [m/s]! If waves are found
to propagate at a lower speed, it would mean that either the water density is different from 1000
[kg/m3] or that the surface is contaminated making s != 0.073 [N/m].
Propagation of Deep-Water Capillary wave
Next, let us consider the case of purely capillary waves on deep water; ie, waves satisfying kh ≤ π
and sk3/ρ >> gk. In this case, the dispersion relation reduces to
σ2 =
sk3
ρ
The wave (phase) speed for the wave is
Cp ≡ σ/k =
√
sk/ρ
while the group speed is
Cg ≡
dσ
dk
=
3
2
√
sk/ρ
In other words, interestingly, the group speed is greater than the phase speed in the case of deep-
water capillary waves. You may recall that in the case of deep-water gravity waves Cg = Cp/2.
32
σ
and the kmin =(ρg/σ)1/2, associated to a wavelength of 1.73 cm for the water, corresponds 
to a minimum for phase velocity (23.2 cm/s).
that shows that there is 
anomalous dispersion
Capillary waves on water have usually wavelengths less than 4mm 
and frequencies higher than 70Hz, thus easily excited by a tuning fork
ω 2 = kg + k 3σ
ρ
⎛
⎝⎜
⎞
⎠⎟
tanh(kh)
ω 2 = k 3σ
ρ
c = σ
ρ
k u = ∂ω
∂ k
= 3
2
c
σ